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Determination of Nonequilibrium Temperature and Pressure using Clausius Equality in a State with Memory: A Simple Model Calculation
P. D. Gujrati

Use of the extended definition of heat dQ=deQ+diQ converts the Clausius inequality dS greater than or equal to deQ/T0 into an equality dS=dQ/T involving the nonequilibrium temperature T of the system having the conventional interpretation that heat flows from hot to cold. The equality is applied to the exact quantum evolution of a 1-dimensional ideal gas free expansion. In a first ever calculation of its kind in an expansion which retains the memory of initial state, we determine the nonequilibrium temperature T and pressure P, which are then compared with the ratio P/T obtained by an independent method to show the consistency of the nonequilibrium formulation. We find that the quantum evolution by itself cannot eliminate the memory effect.cannot eliminate the memory effect; hence, it cannot thermalize the system.

Some Rigorous Results Relating Nonequilibrium, Equilibrium, Calorimetrically Measured and Residual Entropies during Cooling
P. D. Gujrati

We use rigorous nonequilibrium thermodynamic arguments to establish that (i) the nonequilibrium entropy S(T_{0}) of any system is bounded below by the experimentally (calorimetrically) determined entropy S_{expt}(T_{0}), (ii) S_{expt}(T_{0}) is bounded below by the equilibrium or stationary state (such as the supercooled liquid) entropy S_{SCL}(T_{0}) and consequently (iii) S(T_{0}) cannot drop below S_{SCL}(T_{0}). It then follows that the residual entropy S_{R} is bounded below by the extrapolated S_{expt}(0)>S_{SCL}(0) at absolute zero. These results are very general and applicable to all nonequilibrium systems regardless of how far they are from their stationary states.

Nonequilibrium Entropy
P. D. Gujrati

We consider an isolated system in an arbitrary state and provide a general formulation using first principles for an additive and non-negative statistical quantity that is shown to reproduce the equilibrium thermodynamic entropy of the isolated system. We further show that the statistical quantity represents the nonequilibrium thermodynamic entropy when the latter is a state function of nonequilibrium state variables; see text. We consider an isolated 1-d ideal gas and determine its non-equilibrium statistical entropy as a function of the box size as the gas expands freely isoenergetically, and compare it with the equilibrium thermodynamic entropy S_{0eq}. We find that the statistical entropy is less than S_{0eq} in accordance with the second law, as expected. To understand how the statistical entropy is different from thermodynamic entropy of classical continuum models that is known to become negative under certain conditions, we calculate it for a 1-d lattice model and discover that it can be related to the thermodynamic entropy of the continuum 1-d Tonks gas by taking the lattice spacing {\delta} go to zero, but only if the latter is state-independent. We discuss the semi-classical approximation of our entropy and show that the standard quantity S_{f}(t) in the Boltzmann's H-theorem does not directly correspond to the statistical entropy.

Nonequilibrium Thermodynamics. Symmetric and Unique Formulation of the First Law, Statistical Definition of Heat and Work, Adiabatic Theorem and the Fate of the Clausius Inequality: A Microscopic View
P. D. Gujrati

The status of heat and work in nonequilibrium thermodynamics is quite confusing and non-unique at present with conflicting interpretations even after a long history of the first law in terms of exchange heat and work, and is far from settled. Moreover, the exchange quantities lack certain symmetry. By generalizing the traditional concept to also include their time-dependent irreversible components allows us to express the first law in a symmetric form dE(t)= dQ(t)-dW(t) in which dQ(t) and work dW(t) appear on an equal footing and possess the symmetry. We prove that irreversible work turns into irreversible heat. Statistical analysis in terms of microstate probabilities p_{i}(t) uniquely identifies dW(t) as isentropic and dQ(t) as isometric (see text) change in dE(t); such a clear separation does not occur for exchange quantities. Hence, our new formulation of the first law provides tremendous advantages and results in an extremely useful formulation of non-equilibrium thermodynamics, as we have shown recently. We prove that an adiabatic process does not alter p_{i}. All these results remain valid no matter how far the system is out of equilibrium. When the system is in internal equilibrium, dQ(t)\equivT(t)dS(t) in terms of the instantaneous temperature T(t) of the system, which is reminiscent of equilibrium. We demonstrate that p_{i}(t) has a form very different from that in equilibrium. The first and second laws are no longer independent so that we need only one law, which is again reminiscent of equilibrium. The traditional formulas like the Clausius inequality {\oint}d_{e}Q(t)/T_{0}<0, etc. become equalities {\oint}dQ(t)/T(t)\equiv0, etc, a quite remarkable but unexpected result in view of irreversibility. We determine the irreversible components in two simple cases to show the usefulness of our approach; here, the traditional formulation is of no use.

Non-equilibrium thermodynamics. IV: Generalization of Maxwell, Claussius-Clapeyron and Response Functions Relations, and the Prigogine-Defay Ratio for Systems in Internal Equilibrium
P. D. GujratiP. P. Aung

We follow the consequences of internal equilibrium in non-equilibrium systems that has been introduced recently [Phys. Rev. E 81, 051130 (2010)] to obtain the generalization of Maxwell's relation and the Clausius-Clapeyron relation that are normally given for equilibrium systems. The use of Jacobians allow for a more compact way to address the generalized Maxwell relations; the latter are available for any number of internal variables. The Clausius-Clapeyron relation in the subspace of observables show not only the non-equilibrium modification but also the modification due to internal variables that play a dominant role in glasses. Real systems do not directly turn into glasses (GL) that are frozen structures from the supercooled liquid state L; there is an intermediate state (gL) where the internal variables are not frozen. Thus, there is no single glass transition. A system possess several kinds of glass transitions, some conventional (L \rightarrow gL; gL\rightarrow GL) in which the state change continuously and the transition mimics a continuous or second order transition, and some apparent (L\rightarrow gL; L\rightarrow GL) in which the free energies are discontinuous so that the transition appears as a zeroth order transition, as discussed in the text. We evaluate the Prigogine-Defay ratio {\Pi} in the subspace of the observables at these transitions. We find that it is normally different from 1, except at the conventional transition L\rightarrow gL, where {\Pi}=1 regardless of the number of internal variables.

Non-equilibrium thermodynamics.III. Thermodynamic Principles, Entropy Continuity during Component Confinement, Energy Gap and the Residual Entropy
P. D. Gujrati

To investigate the consequences of component confinement such as at a glass transition and the well-known energy or enthalpy gap (between the glass and the perfect crystal at absolute zero, see text), we follow our previous approach [Phys. Rev. E 81, 051130 (2010)] of using the second law applied to an isolated system {\Sigma}_0 consisting of the homogeneous system {\Sigma} and the medium {\Sigma}. We establish on general grounds the continuity of the Gibbs free energy G(t) of {\Sigma} as a function of time at fixed temperature and pressure of the medium. It immediately follows from this and the observed continuity of the enthalpy during component confinement that the entropy S of the open system {\Sigma} must remain continuous during a component confinement such as at a glass transition. We use these continuity properties and the recently developed non-equilibrium thermodynamics to formulate thermodynamic principles of additivity, reproducibility, continuity and stability that must also apply to non-equilibrium systems in internal equilibrium. We find that the irreversibility during a glass transition only justifies the residual entropy S_{R} to be at least as much as that determined by disregarding the irreversibility, a common practice in the field. This justifies a non-zero residual entropy S_{R} in glasses, which is also in accordance with the energy or enthalpy gap at absolute zero. We develop a statistical formulation of the entropy of a non-equilibrium system, which results in the continuity of entropy during component confinement in accordance with the second law and sheds light on the mystery behind the residual entropy, which is consistent with the recent conclusion [Symmetry 2, 1201 (2010)] drawn by us.

Exact Statistical Mechanical Investigation of a Finite Model Protein in its environment: A Small System Paradigm
P. D. GujratiBradley P. Lambeth JrAndrea CorsiEvan Askanazi

We consider a general incompressible finite model protein of size M in its environment, which we represent by a semiflexible copolymer consisting of amino acid residues classified into only two species (H and P, see text) following Lau and Dill. We allow various interactions between chemically unbonded residues in a given sequence and the solvent (water), and exactly enumerate the number of conformations W(E) as a function of the energy E on an infinite lattice under two different conditions: (i) we allow conformations that are restricted to be compact (known as Hamilton walk conformations), and (ii) we allow unrestricted conformations that can also be non-compact. It is easily demonstrated using plausible arguments that our model does not possess any energy gap even though it is supposed to exhibit a sharp folding transition in the thermodynamic limit. The enumeration allows us to investigate exactly the effects of energetics on the native state(s), and the effect of small size on protein thermodynamics and, in particular, on the differences between the microcanonical and canonical ensembles. We find that the canonical entropy is much larger than the microcanonical entropy for finite systems. We investigate the property of self-averaging and conclude that small proteins do not self-average. We also present results that (i) provide some understanding of the energy landscape, and (ii) shed light on the free energy landscape at different temperatures.

Reliably Estimating Bare Chi from Compressible Blends in the Grand Canonical Ensemble
Sagar S. RaneP. D. Gujrati

The bare chi characterizing polymer blends plays a significant role in their macroscopic description. Therefore, its experimental determination, especially from small-angle-neutron-scattering experiments on isotopic blends, is of prime importance in thermodynamic investigations. Experimentally extracted quantity, commonly known as the effective chi is affected by thermodynamics, in particular by polymer connectivity, and density and composition fluctuations. The present work is primarily concerned with studying four possible effective chi's, one of which is closely related to the conventionally defined effective chi, to see which one plays the role of a reliable estimator of the bare chi. We show that the conventionally extracted effective chi is not a good measure of the bare chi in most blends. A related quantity that does not contain any density fluctuations, and one which can be easily extracted, is a good estimator of the bare chi in all blends except weakly interacting asymmetric blends (see text for definition). The density fluctuation contribution is given by (Delta v^bar)**2/2TK_T, where Delta v^bar is the difference of the partial monomer volumes and K_T is the compressibility. Our effective chi's are theory-independent. From our calculations and by explicitly treating experimental data, we show that the effective chi's, as defined here, have weak composition dependence and do not diverge in the composition wings. We elucidate the impact of compressibility and interactions on the behavior of the effective chi's and their relationship with the bare chi.

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